Sequencing of all multinomial coefficients arranged in a s*r array of Pascal simplices P(s,r) and sequenced along the array's anti-diagonals. Each P(s,r) is, in turn, a sequence of terms representing the coefficients of a_1,...,a_s in the expansion of (Sum(a_i, i=1, s))^r with r starting at zero.

The Pascal simplex P(s,r) starts at a(n) where n=2^(s+r-1)+Sum[Binomial[s+r-1,p],{p,0,s-2}]. The individual terms within the Pascal simplex, S(r,t_1,t_2,...,t_(s-1)) are given by S(r,t_1,t_2,...,t_(s-1))=Binomial[r,t_1]*Binomial[t_1,t_2]*...*Binomial[t_(s-2),t_(s-1)].

EXAMPLE

The Pascal simplex P(4,5) for the coefficients of (a_1+a_2+a_3+a_4)^5 is the sequence:-

.......1

.......5

......5,5

.......10

.....20,20

....10,20,10

.......10

.....30,30

....30,60,30

..10,30,30,10

.......5

.....20,20

....30,60,30

..20,60,60,20

..5 ,20,30,20,5

.......1

......5,5

....10,20,10

..10,30,30,10

.5, 20,30,20,5

1,5, 10,10, 5,1

The sequence starts at a(293), it has 56 terms and the sum of its terms is 1024.